Problem Description

You are given an array called nums which only contains positive integers. Your goal is to create the longest possible subset from nums that can fit a specific pattern when arranged in a 0-indexed array. The pattern is such that it forms a palindrome with squares of the base number increasing until the middle of the array, and then decreasing symmetrically. For example, a valid pattern is [x, x^2, x^4, ..., x^(k/2), x^k, x^(k/2), ..., x^4, x^2, x], where x is a base number and k is a non-negative power of 2.

For example, the arrays [2, 4, 16, 4, 2] and [3, 9, 3] meet the criteria for the pattern (palindromic and squares of x), while [2, 4, 8, 4, 2] does not because 8 is not a repeated square of 2 within a palindromic structure. Your task is to return the maximum number of elements in such a subset that meets the described conditions.

Intuition

To solve this problem, we need to analyze the properties of the required pattern. We can see that we can always have at most one number that is not 1 and can be squared multiple times to contribute to the length of our pattern (for example, one instance of 2, two instances of 4, four instances of 16, and so forth). Numbers like 1 can be directly repeated since 1^2 is still 1.

The main observation is that in the valid subset, other than 1, each number can only appear once, and its square can appear at most once too (if present in the original array), forming pairs that will contribute to the pattern symmetrically.

Given such a pattern with a non-repeating core and symmetric structure, a hash table (dictionary in Python) can help us to count the occurrences of each number. By traversing the array and squaring the numbers while they still appear more than once, we determine the maximum length of the subset that can be created.

Numbers that are not 1 will contribute in pairs (except for the center of the pattern if unpaired), whilst 1 can contribute singly but has to be an odd count (or one less than an even count), to ensure that we can place it symmetrically around the center. This way, the code computes the maximum possible length of a subset fitting the given pattern by iterating over each number and its count while updating the maximum length.

Solution Approach

The solution adopts a hash table to keep track of the occurrences of each element in the array nums. In Python, this is done using the Counter class from the collections module, which creates a dictionary-like object that maps elements to their respective counts.

The algorithm proceeds as follows:

• Initialize a counter cnt with the counts of each number in nums.
• Start with calculating the contribution of the number 1. Since ones can be freely placed in the pattern without squaring, you can take all of them if there's an odd count, or one less if there's an even count, to ensure symmetry. This is handled by ans = cnt[1] - (cnt[1] % 2 ^ 1).
• Remove 1 from the counter as it has been handled separately, del cnt[1].

Next, for each unique number x in the counter that is not 1:

• A temporary variable t is introduced to keep track of the contribution of a number to the subset pattern's length.
• While the count of x is more than 1, square x and increase t by 2. This represents adding both x and its square to the subset pattern.
• After exiting the while loop, if there is still one instance of x left, increment t by 1 because it can be used as the central element of the pattern. Otherwise, decrement t by 1 as you've gone one step too far (you've made an even number of selections when you can only have an odd number).
• Update the answer ans with the maximum of the current answer and the tentative contribution t.

The logic of squaring the numbers and checking their counts reflects the idea that numbers in our desired subset can appear either as a single center of the pattern or as pairs flanking the center. Since we are to find the maximum number of elements that can be included, we persistently square the elements and add to our count as long as they can appear in pairs.

After iterating over all numbers in the hash table, we end up with the ans variable holding the maximum possible number of elements that comprise a valid pattern. The result is then returned as the final answer.

Example Walkthrough

Suppose we are given the array nums = [1, 1, 1, 2, 2, 3, 3, 3, 4, 16, 4].

Following the solution approach:

1. Initialize the counter cnt with counts of each number.

   cnt = Counter([1, 1, 1, 2, 2, 3, 3, 3, 4, 16, 4])
   // cnt = {1: 3, 2: 2, 3: 3, 4: 2, 16: 1}

2. Calculate the contribution of number 1:

   ans = 3 - (3 % 2 ^ 1) = 3 - 0 = 3

   We can use all three 1s as they are odd in count and can form a symmetric pattern around the center.

3. Remove 1 from the counter since it's been handled.

   del cnt[1]
   // cnt = {2: 2, 3: 3, 4: 2, 16: 1}

4. Process other numbers starting from the smallest.

   • For 2 in nums:

     • t is initialized to 0.
     • x is now 2, and we have 2 twos.
     • We cannot square 2 to find another 2 in the array, so the contribution of 2 to t remains 0.

   • For 3 in nums:

     • t is initialized to 0.
     • We have three 3s, but we can't square 3 to find another 3 in the array. However, there is an odd count, so we can use one of them as the center. Hence, t is increased by 1.

   • For 4 in nums:

     • t is initialized to 0.
     • We have two 4s. We can square 4, and indeed there is a 16 in the array, which is 4 squared. This means we can add two 4s, and one 16 to the pattern, increasing t by 2 for the 4s, and by 1 for the 16 as the center. Thus, t is increased by 3.

     Update ans with the max of current ans and t:

     ans = max(3, 0 + 3) = 3

   So, our maximum length is 3 from the 1s and 3 from the 4, 16, 4.

5. The algorithm has traversed all numbers in cnt and now ans holds the maximum possible number of elements, which is 6.

Therefore, for nums = [1, 1, 1, 2, 2, 3, 3, 3, 4, 16, 4], the subset [1, 1, 1, 4, 16, 4] reflects the longest subset that can fit the pattern with a length of 6.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is indeed O(n * log log M) where n is the length of the nums array and M is the maximum value in the nums array. This is because we iterate through all elements (O(n)) to populate the Counter, and for each unique number x that is not 1, we may square it repeatedly until it's no longer found in cnt. Squaring a number repeatedly like this takes O(log log x) time for each unique x, assuming x is the maximum value M. Since the squaring process is bounded by the maximum value M, and since it is done for each unique element that is not 1, the complexity is O(n * log log M).

The space complexity is O(n) primarily due to the storage required for the Counter object, which would, in the worst case, store as many elements as are present in the array nums if all elements are unique.

Learn more about how to find time and space complexity quickly using problem constraints.